From: John Hernlund hernlund@............

Date: Mon, 17 Feb 2003 02:29:11 -0800

Hey Chris, In the ideal gas treatment (this is where the approximate comes in),=20= the equation of state is just PV=3DnkT where P is pressure, V is volume,=20= n is number of molecules, k is Boltzman's constant, and T is=20 temperature. This is the same as P=3DrkT where r is the number density.=20= While this is quite approximate, it does capture the gross features of=20= the atmosphere surprisingly well. The hydrostatic part of the=20 atmospheric equilibrium (force balance) is determined by dP/dz=20 proportional to -r*g, where g is the gravitational acceleration. At=20 constant T this gives you: dP/dz proportional to -P*g/kT or upon integrating just P proportional=20 to exp(-z*g/kT) where z is the height. Since P is proportional to r in=20= this case, then density also falls off exponentially with height. The temperature of the atmosphere does decrease slightly up to a=20 certain height, but then rockets way up to thousands of Kelvins in the=20= ionosphere. You can see from the equation of state that at constant=20 pressure, the number density also decreases as temperature increases.=20 At constant temperature, the pressure decreases in proportion to the=20 density. There is obviously a tradeoff between these two end members,=20 which is reflected in the adiabatic (constant entropy) vs isothermal=20 (constant temperature) solutions. But you can certainly see that, no=20 matter what, a decrease in temperature will never lead to a decrease in=20= density unless pressure is increased in larger proportion, which does=20 not occur in the hydrostatic atmosphere. This is all reflected in a=20 term called the "scale height" of the atmosphere, which is the height=20 at which the density decreases by a factor of 1/e =3D 1/2.7182818... = For=20 the Earth's atmosphere this is well measured and is (if my memory suits=20= me well) around 9 kilometers (it is proportional to kT/g from the above=20= considerations). Anyways, this is all old news, and was proposed by Pascal some time=20 ago. The idea was that pressure is primarily due to the weight of the=20 atmosphere above you, so that the hypothesis could be tested by=20 carrying a barometer to the top of a mountain and comparing its reading=20= with the value at the bottom of the mountain. Tests did, of course,=20 confirm this hypothesis. So why all this fuss about density? Density is the amount of mass=20 contained in a volume, and according to Sir Newton if the mass=20 decreases then the acceleration increases given the same forcing. This=20= is what really leads to the large amplitudes for vertically propagating=20= waves... So your supposition about the temperature dependence being most=20 important is not applicable in this case. I would note, however, that=20 it could be appropriate for sound waves traveling horizontally at a=20 constant altitude where the pressure is about constant, and this is=20 probably what you are thinking about. Horizontal temperature changes=20 will then induce density changes in inverse proportion. This is not=20 part of the hydrostatic atmosphere, but rather is part of the dynamic=20 atmosphere which drives winds, etc.. However, this dynamic part is=20 quite small compared to the static part when vertical directions are=20 considered, much like in the deep Earth. The only exception might be=20 when a tornado comes by, and pressure plummets horizontally very=20 rapidly. Also note that the ionosphere is quite a bit higher up there beyond the=20= tropopause, and that the atmosphere does not ever really end but merges=20= in a complicated way with the solar wind and its nature up there is=20 governed mostly by magnetic field effects. Dispersion in the ionosphere=20= is complicated by the polarizing effects of moving cations and anions=20 vertically apart creating a virtual capacitance that damps these=20 oscillations. Cheers! John On Sunday, February 16, 2003, at 06:49 PM, ChrisAtUpw@....... wrote: > In a message dated 16/02/03, hernlund@............ writes: > > Because the atmosphere becomes less dense approximately exponentially=20= > with height, the sound wave velocity decreases. > > Hi John, > > =A0=A0=A0=A0=A0=A0The velocity of sound in air is primarily = proportional to the=20 > square root of the absolute temperature, which of course decreases=20 > with height up to the stratosphere. Pressure has relatively little=20 > effect over the atmospheric range. Dispersion goes roughly with f^2,=20= > where f is the frequency, so waves of a few Hz or less can propagate=20= > around the globe. Volcanic explosions may be detected going around the=20= > Earth several times. > > =A0=A0=A0=A0=A0=A0Regards, > > =A0=A0=A0=A0=A0=A0Chris Chapman= Hey Chris, In the ideal gas treatment (this is where the approximate comes in), the equation of state is just PV=3DnkT where P is pressure, V is volume, n is number of molecules, k is Boltzman's constant, and T is temperature. This is the same as P=3DrkT where r is the number density. While this is quite approximate, it does capture the gross features of the atmosphere surprisingly well. The hydrostatic part of the atmospheric equilibrium (force balance) is determined by dP/dz proportional to -r*g, where g is the gravitational acceleration. At constant T this gives you: dP/dz proportional to -P*g/kT or upon integrating just P proportional to exp(-z*g/kT) where z is the height. Since P is proportional to r in this case, then density also falls off exponentially with height. The temperature of the atmosphere does decrease slightly up to a certain height, but then rockets way up to thousands of Kelvins in the ionosphere. You can see from the equation of state that at constant pressure, the number density also decreases as temperature increases. At constant temperature, the pressure decreases in proportion to the density. There is obviously a tradeoff between these two end members, which is reflected in the adiabatic (constant entropy) vs isothermal (constant temperature) solutions. But you can certainly see that, no matter what, a decrease in temperature will never lead to a decrease in density unless pressure is increased in larger proportion, which does not occur in the hydrostatic atmosphere. This is all reflected in a term called the "scale height" of the atmosphere, which is the height at which the density decreases by a factor of 1/e =3D 1/2.7182818... For the Earth's atmosphere this is well measured and is (if my memory suits me well) around 9 kilometers (it is proportional to kT/g from the above considerations). Anyways, this is all old news, and was proposed by Pascal some time ago. The idea was that pressure is primarily due to the weight of the atmosphere above you, so that the hypothesis could be tested by carrying a barometer to the top of a mountain and comparing its reading with the value at the bottom of the mountain. Tests did, of course, confirm this hypothesis. So why all this fuss about density? Density is the amount of mass contained in a volume, and according to Sir Newton if the mass decreases then the acceleration increases given the same forcing. This is what really leads to the large amplitudes for vertically propagating waves... So your supposition about the temperature dependence being most important is not applicable in this case. I would note, however, that it could be appropriate for sound waves traveling horizontally at a constant altitude where the pressure is about constant, and this is probably what you are thinking about. Horizontal temperature changes will then induce density changes in inverse proportion. This is not part of the hydrostatic atmosphere, but rather is part of the dynamic atmosphere which drives winds, etc.. However, this dynamic part is quite small compared to the static part when vertical directions are considered, much like in the deep Earth. The only exception might be when a tornado comes by, and pressure plummets horizontally very rapidly. Also note that the ionosphere is quite a bit higher up there beyond the tropopause, and that the atmosphere does not ever really end but merges in a complicated way with the solar wind and its nature up there is governed mostly by magnetic field effects. Dispersion in the ionosphere is complicated by the polarizing effects of moving cations and anions vertically apart creating a virtual capacitance that damps these oscillations. Cheers! John On Sunday, February 16, 2003, at 06:49 PM, ChrisAtUpw@....... wrote:= Arial Because the atmosphere becomes less dense approximately exponentially with height, the sound wave velocity decreases. =In a message dated 16/02/03, hernlund@............ writes: Arial 0000,0000,0000Hi John, =A0=A0=A0=A0=A0=A0The velocity of sound in air is primarily proportional = to the square root of the absolute temperature, which of course decreases with height up to the stratosphere. Pressure has relatively little effect over the atmospheric range. Dispersion goes roughly with f^2, where f is the frequency, so waves of a few Hz or less can propagate around the globe. Volcanic explosions may be detected going around the Earth several times. =A0=A0=A0=A0=A0=A0Regards, =A0=A0=A0=A0=A0=A0Chris Chapman